Dr Yann Bernard, Monash University
Elliptic and parabolic partial differential equations are central to the mathematical modelling of physical, geometric, and biological systems. Elliptic equations, such as the Poisson equation, describe steady-state or equilibrium configurations with applications in electrostatics, fluid mechanics, and differential geometry, inter alia. Solutions to elliptic PDEs typically exhibit strong regularity and obey maximum principles, making them analytically tractable and structurally very rich. Parabolic equations, such as the heat equation, govern time-evolving processes involving diffusion and dissipation. These include heat flow, chemical diffusion, population dynamics, and geometric flows like the Ricci and mean curvature flows. Parabolic PDEs tend to smooth out irregularities in initial data over time, revealing long-term behaviour and stability.
The analytical study of partial differential equations was rigorously initiated in the early-to-mid-XXth century and has tremendously grown since then. This introductory course will cover the core results of the theory, and will be theoretical in nature: we will address such questions as existence and uniqueness, but also questions regarding the regularity of solutions, culminating in the resolution of Hilbert’s XIXth problem: are solutions to regular variational problems always analytic?
Week 1: Foundations and Classical Theory of Elliptic Equations
Week 2: Weak Solutions and Sobolev Spaces
Week 3: Linear Parabolic Equations and Schauder Theory
Week 4: Classical Nonlinear Elliptic and Parabolic Equations
Check back for pre-enrolment QUIZ details so you can self-evaluate and get a measure of the key foundational knowledge required.
Yann Bernard completed his doctoral studies at the University of Michigan in 2006. He was a research fellow/postdoc at ETH in Zurich and at the University of Freiburg, before joining the School of Mathematics at Monash University as a senior lecturer in 2015. He currently serves as director of postgraduate studies. His research deals with the study of solutions to conformally invariant variational problems arising in geometric analysis; with applications to elasticity theory, general relativity, and the AdS/CFT correspondence, among others.